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Cherry flow: physical measures and perturbation theory

Published online by Cambridge University Press:  12 May 2016

JIAGANG YANG
JIAGANG YANG*
Affiliation:
Departamento de Geometria, Instituto de Matemática e Estatística, Universidade Federal Fluminense, Niterói, Brazil email [email protected]
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Abstract

In this article we consider Cherry flows on the torus which have two singularities, a source and a saddle, and no periodic orbits. We show that every Cherry flow admits a unique physical measure, whose basin has full volume. This proves a conjecture given by Saghin and Vargas [Invariant measures for Cherry flows. Comm. Math. Phys.317(1) (2013), 55–67]. We also show that the perturbation of Cherry flows depends on the divergence at the saddle: when the divergence is negative, this flow admits a neighborhood, such that any flow in this neighborhood belongs to one of the following three cases: it has a saddle connection; it is a Cherry flow; it is a Morse–Smale flow whose non-wandering set consists of two singularities and one periodic sink. In contrast, when the divergence is non-negative, this flow can be approximated by a non-hyperbolic flow with an arbitrarily large number of periodic sinks.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 8 , December 2017 , pp. 2671 - 2688
Copyright
© Cambridge University Press, 2016 

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References

Alves, J. F., Bonatti, C. and Viana, M.. SRB measures for partially hyperbolic systems whose central direction is mostly expanding. Invent. Math. 140 (2000), 351–398.CrossRefGoogle Scholar
Bonatti, C., Gourmelon, N. and Vivier, T.. Perturbations of the derivative along periodic orbits. Ergod. Th. & Dynam. Sys. 26 1307–1337.CrossRefGoogle Scholar
Bonatti, C. and Viana, M.. SRB measures for partially hyperbolic systems whose central direction is mostly contracting. Israel J. Math. 115 (2000), 157–193.CrossRefGoogle Scholar
Cherry, T.. Analytic quasi-periodic discontinuous type on a torus. Proc. Lond. Math. Soc. 44 (1938), 175–215.CrossRefGoogle Scholar
Gan, S.. A generalized shadowing lemma. Discrete Contin. Dyn. Syst. 8(3) (2002), 627–632.CrossRefGoogle Scholar
Gan, S. and Yang, D.. Morse–Smale systems and horseshoes for three dimensional singular flows. Preprint, 2013, http://arxiv.org/pdf/1302.0946.pdf.Google Scholar
Graczyk, J.. Dynamics of circle maps with flat spots. Fund. Math. 209 (2010), 267–290.CrossRefGoogle Scholar
Graczyk, J., Jonker, L. B., Światek, G., Tangerman, F. M. and Veerman, J. J. P.. Differentiable circle maps with a flat interval. Comm. Math. Phys. 173(3) (1995), 599–622.CrossRefGoogle Scholar
Hu, H. and Young, L. S.. Nonexistence of SRB measures for some diffeomorphisms that are ‘almost Anosov’. Ergod. Th. & Dynam. Sys. 15 (1996), 67–76.CrossRefGoogle Scholar
Katok, A.. Lyapunov exponents, entropy and periodic points of diffeomorphisms. Publ. Math. Inst. Hautes Études Sci. 51 (1980), 137–173.CrossRefGoogle Scholar
Liao, S. T.. An existence theorem for periodic orbits. Acta Sci. Natur. Univ. Pekinensis 1 (1979), 1–20.Google Scholar
Liao, S. T.. A basic property of a certain class of differential systems. Acta Math. Sinica 22 (1979), 316–343 (in Chinese).Google Scholar
Liao, S. T.. Certain uniformity properties of differential systems and a generalization of an existence theorem for periodic orbits. Acta Sci. Natur. Univ. Pekinensis 2 (1985), 1–19.Google Scholar
Liao, S.. On (𝜂, d)-contractible orbits of vector fields. Syst. Sci. Math. Sci. 2 (1989), 193–227.Google Scholar
Liao, S. T.. The Qualitative Theory of Differential Dynamical Systems. Science Press, Beijing, 1996 (translated from the Chinese; distributed by American Mathematical Society, Providence, RI).Google Scholar
Martens, M., van Strien, S., de Melo, W. and Mendes, P.. On Cherry flows. Ergod. Th. & Dynam. Sys. 10 (1990), 531–554.CrossRefGoogle Scholar
Mendes, P.. A metric property of Cherry vector fields on the torus. J. Differential Equations 89(2) (1991), 305–316.CrossRefGoogle Scholar
Moreira, P. C. and Gaspar Ruas, A. A.. Metric properties of Cherry flows. J. Differential Equations 97 (1992), 16–26.CrossRefGoogle Scholar
Palis, J. and de Melo, W.. Geometric Theory of Dynamical Systems, an Introduction. Springer, New York, 1982.CrossRefGoogle Scholar
Palmisano, L.. A phase transition for circle maps and Cherry flows. Comm. Math. Phys. 321(1) (2013), 135–155.CrossRefGoogle Scholar
Palmisano, L.. On physical measures for Cherry flows. Fund. Math. 232 (2016), 167–179.CrossRefGoogle Scholar
Saghin, R., Sun, W. and Vargas, E.. On Dirac physical measures for transitive flows. Comm. Math. Phys. 298(3) (2010), 741–756.CrossRefGoogle Scholar
Saghin, R. and Vargas, E.. Invariant measures for Cherry flows. Comm. Math. Phys. 317(1) (2013), 55–67.CrossRefGoogle Scholar
Viana, M. and Oliveira, K.. Foundations of Ergodic Theory (Cambridge Studies in Advanced Mathematics, 151) . Cambridge University Press, Cambridge, 2015.CrossRefGoogle Scholar
Yang, J.. Topological entropy of Lorenz-like flows. Preprint, 2014, http://arxiv.org/pdf/1412.1207.pdf.Google Scholar